I can't find out much as to what the comparisons  between these two tests re progesterone testing.

Is either of them better or more reliable than the other?

Serum progesterone every time. The in house test kits require careful storage and need bringing to room temperature correctly before use, then there is operator interpretation for colour match on chart. As costs as similar would only use the  in house test as last report over a weekend

Do you know who does the ELISA test KLB? Is it just the repro vets?

Your vet can order them in, but they come in packs of ten tests so most won't have them in stock. 

Many vets will charge for all ten tests as they have few breeders likely to sue them before their use by date.

Any vet can run them, some vets have them available if they have a number of breeders on their books or they will order them in.  Costs much the same as a lab serum progesterone blood. My vet always has then in for weekend checks but they generally recommend lab serum progesterones.